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The mathematical process is as follows. Assume that we conduct M times sam-
pling, then we have cvar = u +
M m =1 [ C m
1
1
1 −α ×
u ] + .
u ] + , Then the problem can be transformed into linear
Suppose t m =[ C m
1
1
form. When λ =
1 −α ×
M ,wehave
cvar = u + λ t m
min
s.t.
t m
0 ,
t m + u
C m
We now discuss when varies λ , how to decide u so that cvar can reach its
minimum.
1. λ =1
(a) when u =max C m , cvar = u =max C m
(b) when u =
M C m =
C + [ C m
C , cvar =
C ] + ,ifwetake c m =
1
max C m ,then cvar > C +max C m
C =max C m , so the circumstance
a is better.
(c) when u =min C m = C 0 , cvar = C 0 + [ C m −C ] + ,ifwetake C m =
max C m ,thenwehave cvar > C 0 +max C m
C 0 =max C m ,inthis
case, circumstance a is still better.
2. λ> 1
This is similar to the situation when λ = 1, we can get that the best u is
max C m through similar analyzing method.
3. λ< 1
Different λ will result in different best u . In this case, we will have to calculate
the best u through linear programming. And the original problem can be
presented,
cvar = u + λ m =1 t m
min
···
u
t 1
.
t M
C 1
1
10
0
C 2
.
10
1
s.t.
.
.
.
10
1
C M
t m
0 ,m =1 ,
···
,M
Local Search. Local search is usually conducted in this way. First, find an
initial solution x . Make use of a simple disturbing method to generate several
adjacent solutions to get a neighborhood. If solution in neighborhood is better
than x , then the initial solution for next local search is this better one, otherwise,
is still x .
In this problem, we generate the neighborhood using the problems property.
When the variance of demand function stays unchanged, based on the results
of simulation, we can know the impact of several intermediate variables to the
decision variables. And the impact is listed below.
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